QMGreensite_merged

25.2. THEQUANTIZATIONOFLIGHT 387

ItsworthstressingthatinHamiltonianformulation,inA 0 = 0 gauge,thecanonical
variablesare{Ai,Ei}, withthe electric field Ei = Pi as the momentum variable
conjugateto the 3-vector potential Ai. Ampere’sLaw follows fromtheHamilton
equationsofmotion

A ̇i(x) = δH

δPi(x,t)

= Ei(x)=Pi(x)

P ̇i(x) = − δH

δAi(x)

= (∇×B%(x))i (25.40)

which,takentogether,give
∂tE%−∇×B%= 0 (25.41)

OftheremainingthreeMaxwellequations,Faraday’sLawand∇·B%= 0 areidentities,
whichfollowfromexpressingE,BintermsofthevectorpotentialA.Theremaining
GaussLaw,∇·E%=0,wehaveagreedtocarryalongseparatelywithoutderivingit
fromtheHamiltonian;thiswasthepriceforimposingA 0 =0.
Nowquantize. TheSchrodinger equationandthemomentum operatorcan be
derived,asbefore,fromEhrenfest’sprinciple

∂t<Ai> = <

δH

δPi(x,t)

>

∂t<Pi> = −<

δH

δAi(x,t)

> (25.42)

buttherearenosurprises,andintheendonefindsthat,justasinordinaryquantum
mechanics

∂tΨ[A] = HΨ[A]

H =

∫

d^3 x[P%^2 +(∇×A%)^2 ]

Pi(x) = −i

δ

δAi(x)

(25.43)

Notethat

[Ai(x),Pj[x′)] = [Ai(x),Ej(x′)]

= iδijδ(x−x′) (25.44)

isthegeneralizationof[x,p]=i ̄hinnon-relativisticquantummechanics.Westillhave
toimposetheGaussLawconstraint,andthisisdonebyimposingthecorresponding
operatorconstraint

(∇·E)Ψ=−i∂i

δ

δAi

Ψ= 0 (25.45)